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<math>X2(t)=\ 2t^2</math>
 
<math>X2(t)=\ 2t^2</math>
  
assume the function <math>Y(t)=\ 5X(t)</math>
+
assume the function <math>Y(t)=\ X(t)</math>
  
<math>Y1(t)=\ 10t</math>
+
<math>Y1(t)=\ 2t</math>
  
<math>Y2(t)=\ 10t^2</math>
+
<math>Y2(t)=\ 2t^2</math>
  
now for <math>aY1(t)+bY2(t)=\ a10t+b10t^2=aX1(t)+bX2(t)</math>
+
now for <math>aY1(t)+bY2(t)=\ a*2t+b*2t^2=[aX1(t)+bX2(t)]</math>
  
thus the given system<math>Y(t)=\ 5X(t)</math>
+
thus the given system<math>Y(t)=\ X(t)</math> is linear
  
  
  
 
==Examples of non linear system==
 
==Examples of non linear system==
 +
<math>X1(t)=\ t</math>
 +
 +
<math>X2(t)=\ t^2</math>
 +
 +
assume the function <math>Y(t)=\ sin[X(t)]</math>
 +
 +
<math>Y1(t)=\ sin(t)</math>
 +
 +
<math>Y2(t)=\ sin(t^2)</math>
 +
 +
now for <math>aY1(t)+bY2(t)=a*sin(t)+b*sin(t^2)\neq sin[aX1(t)+bX2(t)]</math>
 +
 +
thus the given system<math>Y(t)=\ sin[X(t)]</math> is '''not''' a linear system

Latest revision as of 12:33, 11 September 2008

Linear system

A system is said to be linear if it satisfies the principle of superposition i.e if for an input A the system gives an output X and for an input B the system gives output then for an input ( a*A + b*B ) the system should yield the output as ( a*X + b*B ). Where a and b are any complex numbers.

Examples of linear system

$ X1(t)=\ 2t $

$ X2(t)=\ 2t^2 $

assume the function $ Y(t)=\ X(t) $

$ Y1(t)=\ 2t $

$ Y2(t)=\ 2t^2 $

now for $ aY1(t)+bY2(t)=\ a*2t+b*2t^2=[aX1(t)+bX2(t)] $

thus the given system$ Y(t)=\ X(t) $ is linear


Examples of non linear system

$ X1(t)=\ t $

$ X2(t)=\ t^2 $

assume the function $ Y(t)=\ sin[X(t)] $

$ Y1(t)=\ sin(t) $

$ Y2(t)=\ sin(t^2) $

now for $ aY1(t)+bY2(t)=a*sin(t)+b*sin(t^2)\neq sin[aX1(t)+bX2(t)] $

thus the given system$ Y(t)=\ sin[X(t)] $ is not a linear system

Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang